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author Adam Chlipala <adam@chlipala.net>
date Mon, 08 Oct 2012 16:04:49 -0400
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17 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant. Almost every subject covered is also relevant to interactive computer theorem-proving in general, such as for traditional mathematical theorems. In fact, I hope to demonstrate how verified programs are useful as building blocks in all sorts of formalizations. 17 We would all like to have programs check that our programs are correct. Due in no small part to some bold but unfulfilled promises in the history of computer science, today most people who write software, practitioners and academics alike, assume that the costs of formal program verification outweigh the benefits. The purpose of this book is to convince you that the technology of program verification is mature enough today that it makes sense to use it in a support role in many kinds of research projects in computer science. Beyond the convincing, I also want to provide a handbook on practical engineering of certified programs with the Coq proof assistant. Almost every subject covered is also relevant to interactive computer theorem-proving in general, such as for traditional mathematical theorems. In fact, I hope to demonstrate how verified programs are useful as building blocks in all sorts of formalizations.
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19 Research into mechanized theorem proving began around the 1970s, and some of the earliest practical work involved Nqthm%~\cite{Nqthm}\index{Nqthm}%, the "Boyer-Moore Theorem Prover," which was used to prove such theorems as correctness of a complete hardware and software stack%~\cite{Piton}%. ACL2%~\cite{CAR}\index{ACL2}%, Nqthm's successor, has seen significant industry adoption, for instance, by AMD to verify correctness of floating-point division units%~\cite{AMD}%. 19 Research into mechanized theorem proving began in the second half of the 20th century, and some of the earliest practical work involved Nqthm%~\cite{Nqthm}\index{Nqthm}%, the "Boyer-Moore Theorem Prover," which was used to prove such theorems as correctness of a complete hardware and software stack%~\cite{Piton}%. ACL2%~\cite{CAR}\index{ACL2}%, Nqthm's successor, has seen significant industry adoption, for instance, by AMD to verify correctness of floating-point division units%~\cite{AMD}%.
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21 Around the beginning of the 21st century, the pace of progress in practical applications of interactive theorem proving accelerated significantly. Several well-known formal developments have been carried out in Coq, the system that this book deals with. In the realm of pure mathematics, Georges Gonthier built a machine-checked proof of the four color theorem%~\cite{4C}%, a mathematical problem first posed more than a hundred years before, where the only previous proofs had required trusting ad-hoc software to do brute-force checking of key facts. In the realm of program verification, Xavier Leroy led the CompCert project to produce a verified C compiler back-end%~\cite{CompCert}% robust enough to use with real embedded software. 21 Around the beginning of the 21st century, the pace of progress in practical applications of interactive theorem proving accelerated significantly. Several well-known formal developments have been carried out in Coq, the system that this book deals with. In the realm of pure mathematics, Georges Gonthier built a machine-checked proof of the four color theorem%~\cite{4C}%, a mathematical problem first posed more than a hundred years before, where the only previous proofs had required trusting ad-hoc software to do brute-force checking of key facts. In the realm of program verification, Xavier Leroy led the CompCert project to produce a verified C compiler back-end%~\cite{CompCert}% robust enough to use with real embedded software.
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23 Many other recent projects have attracted attention by proving important theorems using computer proof assistant software. For instance, the L4.verified project%~\cite{seL4}% led by Gerwin Klein has given a mechanized proof of correctness for a realistic microkernel, using the Isabelle/HOL proof assistant%~\cite{Isabelle/HOL}\index{Isabelle/HOL}%. The amount of ongoing work in the area is so large that I cannot hope to list all the recent successes, so from this point I will assume that the reader is convinced both that we ought to want machine-checked proofs and that they seem to be feasible to produce. (To readers not yet convinced, I suggest a Web search for "machine-checked proof"!) 23 Many other recent projects have attracted attention by proving important theorems using computer proof assistant software. For instance, the L4.verified project%~\cite{seL4}% led by Gerwin Klein has given a mechanized proof of correctness for a realistic microkernel, using the Isabelle/HOL proof assistant%~\cite{Isabelle/HOL}\index{Isabelle/HOL}%. The amount of ongoing work in the area is so large that I cannot hope to list all the recent successes, so from this point I will assume that the reader is convinced both that we ought to want machine-checked proofs and that they seem to be feasible to produce. (To readers not yet convinced, I suggest a Web search for "machine-checked proof"!)
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128 (** * Engineering with a Proof Assistant *) 128 (** * Engineering with a Proof Assistant *)
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131 In comparisons with its competitors, Coq is often derided for promoting unreadable proofs. It is very easy to write proof scripts that manipulate proof goals imperatively, with no structure to aid readers. Such developments are nightmares to maintain, and they certainly do not manage to convey "why the theorem is true" to anyone but the original author. One additional (and not insignificant) purpose of this book is to show why it is unfair and unproductive to dismiss Coq based on the existence of such developments. 131 In comparisons with its competitors, Coq is often derided for promoting unreadable proofs. It is very easy to write proof scripts that manipulate proof goals imperatively, with no structure to aid readers. Such developments are nightmares to maintain, and they certainly do not manage to convey "why the theorem is true" to anyone but the original author. One additional (and not insignificant) purpose of this book is to show why it is unfair and unproductive to dismiss Coq based on the existence of such developments.
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133 I will go out on a limb and guess that the reader is a fan of some programming language, and that he may even have been involved in teaching that language to undergraduates. I want to propose an analogy between two attitudes: coming to a negative conclusion about Coq after reading common Coq developments in the wild, and coming to a negative conclusion about Your Favorite Language after looking at the programs undergraduates write in it in the first week of class. The pragmatics of mechanized proving and program verification have been under serious study for much less time than the pragmatics of programming have been. The computer theorem proving community is still developing the key insights that correspond to those that programming texts and instructors impart to their students, to help those students get over that critical hump where using the language stops being more trouble than it is worth. Most of the insights for Coq are barely even disseminated among the experts, let alone set down in a tutorial form. I hope to use this book to go a long way towards remedying that. 133 I will go out on a limb and guess that the reader is a fan of some programming language and may even have been involved in teaching that language to undergraduates. I want to propose an analogy between two attitudes: coming to a negative conclusion about Coq after reading common Coq developments in the wild, and coming to a negative conclusion about Your Favorite Language after looking at the programs undergraduates write in it in the first week of class. The pragmatics of mechanized proving and program verification have been under serious study for much less time than the pragmatics of programming have been. The computer theorem proving community is still developing the key insights that correspond to those that programming texts and instructors impart to their students, to help those students get over that critical hump where using the language stops being more trouble than it is worth. Most of the insights for Coq are barely even disseminated among the experts, let alone set down in a tutorial form. I hope to use this book to go a long way towards remedying that.
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135 If I do that job well, then this book should be of interest even to people who have participated in classes or tutorials specifically about Coq. The book should even be useful to people who have been using Coq for years but who are mystified when their Coq developments prove impenetrable by colleagues. The crucial angle in this book is that there are "design patterns" for reliably avoiding the really grungy parts of theorem proving, and consistent use of these patterns can get you over the hump to the point where it is worth your while to use Coq to prove your theorems and certify your programs, even if formal verification is not your main concern in a project. We will follow this theme by pursuing two main methods for replacing manual proofs with more understandable artifacts: dependently typed functions and custom Ltac decision procedures. 135 If I do that job well, then this book should be of interest even to people who have participated in classes or tutorials specifically about Coq. The book should even be useful to people who have been using Coq for years but who are mystified when their Coq developments prove impenetrable by colleagues. The crucial angle in this book is that there are "design patterns" for reliably avoiding the really grungy parts of theorem proving, and consistent use of these patterns can get you over the hump to the point where it is worth your while to use Coq to prove your theorems and certify your programs, even if formal verification is not your main concern in a project. We will follow this theme by pursuing two main methods for replacing manual proofs with more understandable artifacts: dependently typed functions and custom Ltac decision procedures.
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158 There, you can find all of the code appearing in this book, with prose interspersed in comments, in exactly the order that you find in this document. You can step through the code interactively with your chosen graphical Coq interface. The code also has special comments indicating which parts of the chapters make suitable starting points for interactive class sessions, where the class works together to construct the programs and proofs. The included Makefile has a target <<templates>> for building a fresh set of class template files automatically from the book source. 158 There, you can find all of the code appearing in this book, with prose interspersed in comments, in exactly the order that you find in this document. You can step through the code interactively with your chosen graphical Coq interface. The code also has special comments indicating which parts of the chapters make suitable starting points for interactive class sessions, where the class works together to construct the programs and proofs. The included Makefile has a target <<templates>> for building a fresh set of class template files automatically from the book source.
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160 A traditional printed version of the book is slated to appear from MIT Press in the future. The online versions will remain available at no cost even after the printed book is released, and I intend to keep the source code up-to-date with bug fixes and compatibility changes to track new Coq releases. 160 A traditional printed version of the book is slated to appear from MIT Press in the future. The online versions will remain available at no cost even after the printed book is released, and I intend to keep the source code up-to-date with bug fixes and compatibility changes to track new Coq releases.
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162 %\index{graphical interfaces to Coq}%I believe that a good graphical interface to Coq is crucial for using it productively. I use the %\index{Proof General}%{{http://proofgeneral.inf.ed.ac.uk/}Proof General} mode for Emacs, which supports a number of other proof assistants besides Coq. There is also the standalone %\index{CoqIDE}%CoqIDE program developed by the Coq team. I like being able to combine certified programming and proving with other kinds of work inside the same full-featured editor, and CoqIDE has had a good number of crashes and other annoying bugs in recent history, though I hear that it is improving. In the initial part of this book, I will reference Proof General procedures explicitly, in introducing how to use Coq, but most of the book will be interface-agnostic, so feel free to use CoqIDE if you prefer it. The one issue with CoqIDE, regarding running through the book source, is that I will sometimes begin a proof attempt but cancel it with the Coq [Abort] or #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Restart%}%#</span></span># commands, which CoqIDE does not support. It would be bad form to leave such commands lying around in a real, finished development, but I find these commands helpful in writing single source files that trace a user's thought process in designing a proof. 162 %\index{graphical interfaces to Coq}%I believe that a good graphical interface to Coq is crucial for using it productively. I use the %\index{Proof General}%{{http://proofgeneral.inf.ed.ac.uk/}Proof General} mode for Emacs, which supports a number of other proof assistants besides Coq. There is also the standalone %\index{CoqIDE}%CoqIDE program developed by the Coq team. I like being able to combine certified programming and proving with other kinds of work inside the same full-featured editor. In the initial part of this book, I will reference Proof General procedures explicitly, in introducing how to use Coq, but most of the book will be interface-agnostic, so feel free to use CoqIDE if you prefer it. The one issue with CoqIDE, regarding running through the book source, is that I will sometimes begin a proof attempt but cancel it with the Coq [Abort] or #<span class="inlinecode"><span class="id" type="keyword">#%\coqdockw{%Restart%}%#</span></span># commands, which CoqIDE does not support. It would be bad form to leave such commands lying around in a real, finished development, but I find these commands helpful in writing single source files that trace a user's thought process in designing a proof.
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165 (** ** Reading This Book *) 165 (** ** Reading This Book *)
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168 For experts in functional programming or formal methods, learning to use Coq is not hard, in a sense. The Coq manual%~\cite{CoqManual}% and the textbook by Bertot and Cast%\'%eran%~\cite{CoqArt}% have helped many people become productive Coq users. However, I believe that the best ways to manage significant Coq developments are far from settled. In this book, I mean to propose my own techniques, and, rather than treating them as advanced material for a final chapter or two, I employ them from the very beginning. After a first chapter showing off what can be done with dependent types, I retreat into simpler programming styles for the first part of the book. I adopt the other main thrust of the book, Ltac proof automation, more or less from the very start of the technical exposition. 168 For experts in functional programming or formal methods, learning to use Coq is not hard, in a sense. The Coq manual%~\cite{CoqManual}%, the textbook by Bertot and Cast%\'%eran%~\cite{CoqArt}%, and Pierce et al.'s %\emph{%Software Foundations%}\footnote{\url{http://www.cis.upenn.edu/~bcpierce/sf/}}% have helped many people become productive Coq users. However, I believe that the best ways to manage significant Coq developments are far from settled. In this book, I mean to propose my own techniques, and, rather than treating them as advanced material for a final chapter or two, I employ them from the very beginning. After a first chapter showing off what can be done with dependent types, I retreat into simpler programming styles for the first part of the book. I adopt the other main thrust of the book, Ltac proof automation, more or less from the very start of the technical exposition.
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170 Some readers have suggested that I give multiple recommended reading orders in this introduction, targeted at people with different levels of Coq expertise. It is certainly true that Part I of the book devotes significant space to basic concepts that most Coq users already know quite well. However, as I am introducing these concepts, I am also developing my preferred automated proof style, so I think even the chapters on basics are worth reading for experienced Coq hackers. 170 Some readers have suggested that I give multiple recommended reading orders in this introduction, targeted at people with different levels of Coq expertise. It is certainly true that Part I of the book devotes significant space to basic concepts that most Coq users already know quite well. However, as I am introducing these concepts, I am also developing my preferred automated proof style, so I think even the chapters on basics are worth reading for experienced Coq hackers.
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172 Readers with no prior Coq experience can ignore the preceding discussion! I hope that my heavy reliance on proof automation early on will seem like the most natural way to go, such that you may wonder why others are spending so much time entering sequences of proof steps manually. 172 Readers with no prior Coq experience can ignore the preceding discussion! I hope that my heavy reliance on proof automation early on will seem like the most natural way to go, such that you may wonder why others are spending so much time entering sequences of proof steps manually.
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174 Coq is a very complex system, with many different commands driven more by pragmatic concerns than by any overarching aesthetic principle. When I use some construct for the first time, I try to give a one-sentence intuition for what it accomplishes, but I leave the details to the Coq reference manual%~\cite{CoqManual}%. I expect that readers interested in complete understandings will be consulting that manual frequently; in that sense, this book is not meant to be completely standalone. I often use constructs in code snippets without first introducing them at all, but explanations should always follow in the prose paragraphs immediately after the offending snippets. 174 Coq is a very complex system, with many different commands driven more by pragmatic concerns than by any overarching aesthetic principle. When I use some construct for the first time, I try to give a one-sentence intuition for what it accomplishes, but I leave the details to the Coq reference manual%~\cite{CoqManual}%. I expect that readers interested in complete understanding will be consulting that manual frequently; in that sense, this book is not meant to be completely standalone. I often use constructs in code snippets without first introducing them at all, but explanations should always follow in the prose paragraphs immediately after the offending snippets.
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176 Previous versions of the book included some suggested exercises at the ends of chapters. Since then, I have decided to remove the exercises and focus on the main book exposition. Especially in the domain of interactive theorem proving, keeping exercises at the proper difficulty level is a lot of work, I have learned! My advice to readers is to learn to use Coq by getting started applying it in their own real projects, rather than focusing on artificial exercises. 176 Previous versions of the book included some suggested exercises at the ends of chapters. Since then, I have decided to remove the exercises and focus on the main book exposition. Especially in the domain of interactive theorem proving, keeping exercises at the proper difficulty level is a lot of work, I have learned! My advice to readers is to learn to use Coq by getting started applying it in their own real projects, rather than focusing on artificial exercises.
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179 (** ** On the Tactic Library *) 179 (** ** On the Tactic Library *)